Not every possible set. All you need is a primitive recursive procedure to decide whether a given statement is an axiom, and primitive recursive procedures to check whether each rule of inference is correctly followed in going from one step to another.
Right: automated proof checking is not really “generally doable” in an axiomatic system—how hard it is depends entirely on the axiomatic system under discussion.
Not every possible set. All you need is a primitive recursive procedure to decide whether a given statement is an axiom, and primitive recursive procedures to check whether each rule of inference is correctly followed in going from one step to another.
Right: automated proof checking is not really “generally doable” in an axiomatic system—how hard it is depends entirely on the axiomatic system under discussion.